Anisotropic elastic behaviour using the four-dimensional formalism of differential geometry

This paper discusses covariance and material objectivity in continuum mechanics. The aim is to extend the mathematical framework of constitutive relations to the four-dimensional (4D) formalism of the General Relativity theory. First, it is demonstrated that 4D general linear or non-linear, isotropic or anisotropic relations can be obtained, no matter the reference frame, such as inertial, animated with a rigid body motion and convective or generally curvilinear, and no matter the starting point, for instance a variation of a thermodynamic potential or a direct coupling between stress-like and strain-like tensors, etc. Second, it is then always possible to model anisotropic behaviour, which is not always possible in some of the 3D classical approaches. In order to demonstrate this 4D approach, different elastic relations for different observers have been investigated analytically and numerically.